An active cloaking strategy for the scalar Helmholtz equation in three dimensions is developed by placing active sources at the vertices of Platonic solids. In each case, a “silent zone” is created interior to the Platonic solid and only the incident field remains in a defined region exterior to this zone. This distribution of sources ensures that implementation of the cloaking strategy is efficient: once the multipole source amplitudes at a single source location are determined, the other amplitudes are calculated by multiplying the multipole source vector by a rotation matrix. The technique is relevant to any scalar wave field.
Over the last two decades, excitement has been generated around the idea of rendering objects invisible to incident excitation (Miller, 2006; Norris, 2008; Schurig , 2006), via passive (Cai , 2007; Silveirinha , 2007; Zhang , 2011) or active (Miller, 2006; Vasquez , 2009a; Zheng , 2010) cloaking. The latter employs sources to suppress fields and has been of interest since Lueg's patent in 1936 (Guicking, 1990). Significant work on anti-sound/anti-vibration has been carried out since then (Cheer, 2016; Fuller , 1996; Guicking, 2007; Nelson and Elliott, 1991). Specific choices for the active sources ensure quiet zones and illusions (Ma , 2013; Zheng , 2010) and control arrangements can be optimised to reduce acoustic scattering in a stationary fluid (Eggler , 2019a; House , 2020; Lin , 2021; O'Neill , 2015) and in a convected flow field (Eggler , 2019b).
Recent interest has centred on object-independent active cloaking methods, e.g., Miller's sensing method (Miller, 2006). This approach cannot provide a relationship between the incident field and source amplitudes however, so this was addressed in (Vasquez , 2009a,b, 2011), using multipole sources to create silent zones in two dimensions (2D). Further progress came in (Norris , 2012), where closed-form explicit formulas for the active source coefficients were deduced. This was also extended to elastodynamics (Futhazar , 2015; Norris , 2014). This approach is attractive because the active source coefficients are independent of the object to be cloaked. This is in contrast to object-dependent approaches, which must be modified in resonant regimes (O'Neill , 2016). The latter approach, however, does not suffer from large amplitudes in the vicinity of the active source regions.
Although there has been extensive work on active manipulation of sound in three dimensions (3D) (Ahrens, 2012; Egarguin , 2020; Elliott , 2012; Onofrei and Platt, 2018), the object-independent approach has thus far been conducted in 2 D only, with the sole exception of Vasquez (2013), which considered the Helmholtz equation in 3D, where results were provided for the case of four active sources.
Here, we provide a framework for active cloaking in 3D. We derive expressions for the active source coefficients and introduce a fast, efficient methodology by placing active sources on vertices of the Platonic solids and exploiting their symmetry. Regardless of the incident field, once active source coefficients have been deduced for one of the active sources, the remaining source amplitudes can subsequently be determined, thus providing a complete exposition of 3D active cloaking for the Helmholtz equation.
We consider active exterior cloaking for time-harmonic waves (with dependence , where ω is the angular frequency and t is time) governed by the 3D, homogeneous Helmholtz equation , where is the wavenumber, with c the wavespeed. For acoustics, the scalar field is the velocity potential at .
To achieve active cloaking, are sought such that for some closed domain C surrounded by the active sources with , we have for and as . Thus, while the scattered field from any object is nullified in C, we also stipulate that the radiation of ud itself to the far field is zero, leaving minimal evidence of the cloak.
The active sources are located at the vertices of an imaginary Platonic solid, thus limiting the number of sources L to five values (Euclid, 2012) [see Figs. 2(a)–2(e)]. This geometry ensures that the sources reside on the circumsphere of the Platonic solids such that for all in each case, where x0 is an arbitrary constant. We set in every case such that the active source with always has the lowest z-coordinate. To locate the remaining sources, we note that a Platonic solid consisting of q p-sided regular polygonal faces around each vertex can be characterized by two indices (p, q) [e.g., a cube with three squares around each vertex has the indices (4, 3)]. We define the length of each side of a Platonic solid as sx0.
|L .||(p, q) .||s .||(in units of ) .|
|L .||(p, q) .||s .||(in units of ) .|
In Figs. 2(a)–2(e), we illustrate the respective cloaked regions inside each of the Platonic solids with a taken as the lower bound in Eq. (10) (when the volume of the domain is a maximum). In each case, the surface of integration is delimited by q identical circular arcs (q depends on L as indicated in Table 1). For non-Platonic distributions of sources, these arcs are not identical.
The method is illustrated in Fig. 4, with L = 20 active sources distributed at the vertices of a regular dodecahedron with source distance , which was produced using a Mathematica code that is available to download from https://github.com/himyeung1025/3d_silent_zone_cloaking. We consider the problem in the context of acoustics, but the principles are similar for other scalar waves. In Figs. 4(a)–4(d), the incident plane wave propagates in the positive x direction with ; in Fig. 4(e)–4(h), it has angles of incidence . In Figs. 4(a), 4(b), 4(e), and 4(f), the scattering object inside the cloaked region C is a sound-soft sphere with radius , where , and a is chosen to be the minimum permissible radius of the imaginary spheres bounding C for L = 4, which means that the scattering sphere has a radius three times that of the inscribed sphere of C when L = 4. In Figs. 4(c), 4(d), 4(g), and 4(h), we show the case where there is no scattering object. In all subplots, the real part of the total wave field u on the cross-section z = 0 is shown. In Figs. 4(a), 4(c), 4(e), and 4(g), the cloaking devices are inactive and a prominent scattering pattern, including distorted wavefronts and a shadow region behind the sphere, is observed. In Figs. 4(b), 4(d), 4(f), and 4(h), the cloaking devices are activated and the multipole order of each active source is taken as N = 10. The series expansions for the source coefficients in Eq. (11) are truncated such that the active field produced by each source is within 1% relative error. The spheres now reside in quasi-silent regions. The straighter wavefronts and the absence of the shadow region in the figures indicate that the incident wave is scattered only slightly by the sphere inside the quiet zone and the sources radiate little into the exterior of the silent region, demonstrating the cloak's effectiveness. The wave field diverges within small neighbourhoods of the active point sources. In practice, these large fields are confined within the finite-sized sources. We have cropped the plot at .
For each kA, the truncation parameter is chosen to achieve a relative error of less than 1% in the active field generated by each source. We require and thus . For all cases, the SWL remains negative for the range of kA studied here with increased efficacy at lower frequencies. As the source number L increases the cloaking effect generally improves, with a maximum reduction of approximately 70 decibels achieved when L = 20 at . Further parameter studies can be found in Part 6 of the supplementary material.1
In conclusion, we have formulated an efficient, 3D, active exterior cloaking strategy for the scalar Helmholtz equation. The Platonic distribution of the active sources means that we only need to determine the source amplitudes at one location; those at other locations follow from post-processing, exploiting the symmetry and regularity of the Platonic solids.
This work was supported by a University of Manchester President's scholarship for Yeung (2017-21) and by the Engineering and Physical Sciences Research Council (EP/L018039/1) for Parnell. The authors have no conflicts of interest to declare. The data in this paper was generated using Mathematica code that is available to download at https://github.com/himyeung1025/3d_silent_zone_cloaking.
See supplementary material at https://doi.org/10.1121/10.0019906 for detailed mathematical derivations of the active noise control model.